# Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography

arXiv: https://arxiv.org/abs/2609.04165 · HTML: https://mlsanity.com/paper/2609.04165/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 115 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [Adaptive Learning of Tensor Network Structures](https://mlsanity.com/paper/2008.05437/) (2020)
- [An Improved Parameterized Algorithm for Treewidth](https://mlsanity.com/paper/2211.07154/) (2022)
- [Hamiltonian Property Testing](https://mlsanity.com/paper/2403.02968/) (2024)
- [Efficient Contraction of Large Tensor Networks for Weighted Model Counting through Graph Decompositions](https://mlsanity.com/paper/1908.04381/) (2019)
- [Tolerant Testing of Stabilizer States with Mixed State Inputs](https://mlsanity.com/paper/2411.08765/) (2024)
- [Testing classical properties from quantum data](https://mlsanity.com/paper/2411.12730/) (2024)
- [Mildly-Interacting Fermionic Unitaries are Efficiently Learnable](https://mlsanity.com/paper/2504.11318/) (2025)
- [Certifying and learning quantum Ising Hamiltonians](https://mlsanity.com/paper/2509.10239/) (2025)
- [Optimal lower bounds for quantum state tomography](https://mlsanity.com/paper/2510.07699/) (2025)
- [Learning the structure of open quantum systems](https://mlsanity.com/paper/2606.30358/) (2026)
- [Efficient quantum tomography](https://mlsanity.com/paper/1508.01907/) (2015)
- [Efficient learning of bosonic Gaussian unitaries](https://mlsanity.com/paper/2510.05531/) (2025)
- [Breaking the Treewidth Barrier in Quantum Circuit Simulation with Decision Diagrams](https://mlsanity.com/paper/2510.06775/) (2025)
- [Optimal certification of constant-local Hamiltonians](https://mlsanity.com/paper/2512.09778/) (2025)

## Abstract (arXiv)

Parameterised graph theory studies how the complexity of graph-theoretic problems depends on structural parameters of the input graph. This perspective has proved useful in analysing tensor-network simulation (Markov and Shi, 2008). Its implications for tensor-network representations and tomography are less well understood. In particular, which graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state? We address these questions using parameterised graph theory. First, we show that cutwidth and tree-cutwidth bound the bond dimension overhead required to represent a TNS as an MPS or TTN. In the TTN case, tree-cutwidth also bounds the local dimension of the grouped subsystems. The proofs are based on entanglement rerouting, a tensor-network analogue of rerouting information in a classical network. Second, we derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth. We obtain these results by extending the disentangling MPS learner of (Cramer et al., 2010), as analysed further in (Bakshi et al., 2025; Lin et al., 2025), to TTNs and to tensor networks on arbitrary known graphs. Finally, we extend the framework beyond the realisable setting. For an arbitrary input state, our agnostic learner outputs a pure state whose fidelity is within additive error ε of the optimum over tensor-network states on the given graph with a given bond dimension, with explicit graph-dependent bounds on sample and computational complexity.